FFT vs Filon's Rule: Accuracy Comparison

[chafelix]

Homework Statement

Suppose we can calculate a quantity f(t) and we need its Fourier transform F(w). Looks to me that accuracywise Filon's rule, e.g. approximating the computed f(t) by splines and analytically integrating piecewise should be more accurate than an FFT, at least for a smooth f(t); especially for large enough w(>w_Nyquist) the FFT would show aliasing, while Filon's rule might lose accuracy due to the summation of a large number of terms of possibly different sign. Why is FFT always used?

 

[Valkarie]

Homework Equations N/AThe Attempt at a Solution FFT is used because it is more computationally efficient than Filon's rule. While Filon's rule requires repeated integrations of the approximated spline, FFT is just a sequence of fast discrete Fourier transforms that can be implemented using matrix multiplication algorithms. Additionally, FFT can be used to calculate the Fourier transform of a function of any length and resolution, whereas Filon's rule is only applicable to evenly spaced data points.